 
Summary: FROM ITERATIVE ALGEBRAS TO ITERATIVE THEORIES
JI
R I AD
AMEK, STEFAN MILIUS, AND JI
R I VELEBIL
Abstract. Iterative theories introduced by Calvin Elgot formalize potentially innite computations as
solutions of recursive equations. One of the main results of Elgot and his coauthors is a description of
a free iterative theory as the theory of all rational trees. Their algebraic proof of this fact is extremely
complicated. In our paper we show that by starting with \iterative algebras", i. e., algebras admitting a
unique solution of all systems of
at recursive equations, a free iterative theory is obtained as the theory of
free iterative algebras. The (coalgebraic) proof we present is dramatically simpler than the original algebraic
one. And our result is, nevertheless, much more general: we describe a free iterative theory on any nitary
endofunctor of every locally presentable category A. This allows us, e. g., to consider iterative algebras over
any equationally specied class A of nitary algebras.
Reportedly, a blow from the welterweight boxer
Norman Selby, also known as Kid McCoy, left one
victim proclaiming, \It's the real McCoy!"
[TPT]
1. Introduction
Iterative theories have been introduced by Calvin C. Elgot [E] as a model of computation formalized as a
